Graph Pooling GNNs represent an emerging trend in AI due to their natural inclination for learning from structured data.
A recent research direction in the domain of graph classification regards learning valuable node clusters in the pooling stage, i.e., hierarchical learning. This process can be informally thought as decomposing a graph into a set of distinct node groups, which are viewed as its building blocks. The first example of hierarchical learning is given in [1]}, where a differentiable node clustering layer, labelled as DiffPool, defines a high-level graph representation via a soft node assignment operation. Subsequently, several node clustering architectures have been proposed in order to learn efficient node clusters while avoiding degenerate scenarios, such as uniform node clusters [2]} or smooth node representations [3]}. Notable examples are given by SAGPool [4]}, minCutPool [2]}, HGP-SL [6]} and SOPool [7]}. Differently from these approaches, we propose a simple solution for structured learning subdued to domain specific constraints, by applying regularizations at the pooling layer level. In particular, we do not require any modification of the model architecture, provided that node pooling is defined as a soft assignment operation. In this way, it is possible to encode high-demanding requirements by defining structured regularizations in a plug-and-play fashion.
Clearly, it is also possible to directly encode the described regularization functions within the graph convolution operation, by defining more powerful graph architectures such as in [8]}, [9]}. In particular, our method defines a soft approximation of a TK, that is, the network learns to extract the \(k\)  most important tree fragments. A graph convolution based formulation would consider all tree fragments in the node encoding operation. In this case, the aggregation function (Eq. REF ) views node groups as neighbours in compliance with a given TK. To ensure efficiency, a hierarchical formulation as in [8]} might be considered. Yet, such method directly affects the network architecture and its formulation might be more complex than a set of soft regularization functions. On the other hand, the soft nature of such constraints inherently gives space to degenerate scenarios that, in order to be avoided, might require additional penalties and dynamic optimization methods.
